Routeburn in summer - Boots or shoes?

I think you will be pretty much right in a good pair of shoes. Most of the track is of a very high standard and there aren't many places left on the track where you are likely to get wet. The number one most important thing with footwear when tramping in my mind is comfort. If shoes are all you can wear then that's what you should wear. Off track your feet may need a bit more protection but there is no need for great leather clod hoppers. There are some quite nice light boots around now. They are much better than they used to be.